Suppose that R (red) and B (blue) are two graphs on the same vertex set of size n, and H is some graph with a red-blue coloring of its edges. How large can R and B be if R∪ B does not contain a copy of H? Call the largest such integer mex(n, H). This problem was introduced by Diwan and Mubayi, who conjectured that (except for a few specific exceptions) when H is a complete graph on k+1 vertices with any coloring of its edges mex(n,H)=ex(n, K_k+1). This conjecture generalizes Turán's theorem.

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... an and Mubayi also asked for an analogue of Erdős-Stone-Simonovits theorem in this context. We prove the following asymptotic characterization of the extremal threshold in terms of the chromatic number χ(H) and the reduced maximum matching number ℳ(H) of H. mex(n, H)=(1- 1/2(χ(H)-1) - Ω(ℳ(H)/χ(H)^2))n^2/2. ℳ(H) is, among the set of proper χ(H)-colorings of H, the largest set of disjoint pairs of color classes where each pair is connected by edges of just a single color. The result is also proved for more than 2 colors and is tight up to the implied constant factor. We also study mex(n, H) when H is a cycle with a red-blue coloring of its edges, and we show that mex(n, H)≲1/2n2, which is tight.